
This flow equation has been obtained first by Cardy in [1]} from random geometry considerations, which serves as a consistency check for our result.
We can also apply the flow equation to the closed channel partition function. We define \(F_{ab}^{\phi }(R)=(G_a^{\phi }(R))^*G_b^{\phi }(R)\) , identifying each Boltzmann weight, we have
\(\partial _{\lambda }\left(F_{ab}^{\phi }(R,\lambda )e^{-E_{\phi }(R,\lambda )L}\right)=-\left(\frac{\partial }{\partial R}-\frac{1}{R}\right)\partial _L\left(F_{ab}^{\phi }(R,\lambda )e^{-E_{\phi }(R,\lambda )L} \right).\) 
